Abstract
We introduce the Szász and Chlodowsky operators based on Gould-Hopper polynomials and study the statistical convergence of these operators in a weighted space of functions on a positive semiaxis. Further, a Voronovskaja type result is obtained for the operators containing Gould-Hopper polynomials. Finally, some graphical examples for the convergence of this type of operator are given.
1. Introduction
In [1], Jakimovski and Leviatan introduced and investigated some approximation properties of the Favard-Szász type operator, by using Appell polynomials which satisfy the identitywhere is an analytic function in the disc , and ,Varma et al. constituted a link between orthogonal polynomials and the positive linear operators. In [1], they proposed Szász operators involving the Brenke polynomials defined by where and
Recently, Büyükyazıcı et al. in [2] introduced the Chlodowsky variant of operators (3). Inspired by this work, we give the Szász-Chlodowsky type operators including Gould-Hopper polynomials. The generating functions for these Gould-Hopper polynomials are given byand the explicit representationswhere denotes the integer part. Now, let us define Szász-Chlodowsky type generalization of the Szász operators with the help of generating function (4), as follows:where and is a positive increasing sequence with the propertiesThe generalization of Szász type operators has been studied in [3–10].
The aim of the this paper is to study some direct results in terms of the modulus of continuity of the second order, convergence of derivative operators to derivative functions, the weighted space, and the degree of approximation of by . We also study the statistical convergence. The rate of convergence of the operators to a certain function is also illustrated through graphics using Matlab.
2. Notations and Auxiliary Results
Let us denote , for some The following notations and lemmas are needed to prove the main results.
In what follows, let , , be the test functions.
Lemma 1. From (4), one has (i);(ii);(iii);(iv);(v) + .
Lemma 2. For the operators , one has (i);(ii);(iii);(iv);(v) + .
Proof. From Lemma 2 and by definition of , we haveNow, we consider the case as follows: For , we have and, finally,
Lemma 3. Let , and then, for every , one has (i);(ii);(iii)
Theorem 4. Let Then, , uniformly on each compact subset of
Proof. By Lemma 2, , , uniformly on every compact subset of So, by Bohman-Korovkin theorem, the result follows.
Example 5. For , , and , , and , the convergence of to is illustrated in Figure 1(a).
(a)
(b)
Example 6. For , , and , , and , the convergence of to is illustrated in Figure 1(b).
Theorem 7. Let , and then for any one has
Proof. By using Lemma 3 case (ii) and the well-known properties of the modulus of continuity, we haveRecalling the Cauchy-Schwarz inequality, we obtain the formula below:By means of Lemma 3 case (ii), for , one gets Using (22) and taking in (19), we obtain the desired result.
The theorem below shows that the derivative is also an approximation process for
Theorem 8. Let If exists at a point , then one has where is the modulus of continuity of
Proof. By simple calculations, the following formula is obtained:where is the difference of order of corresponding to the increment . Using the relation between finite difference and divided difference, the derivative of order of the operators is represented as follows: where Then, using Theorem 4, we haveBy using the mean value theorem and some known classical properties of the modulus of continuity [11], one has where Hence, we obtain On the other hand, where Using the estimates in (15), we have the desired result.
3. Weighted Approximation
Let be the space of all functions defined on satisfying the condition , where is a positive constant depending only on and is a weight function. By , we denote the subspace of all continuous functions with the norm and
Theorem 9. Let , and let be a weighted function; then, the inequality is satisfied.
Proof. Using Lemma 2, one has Since , there exists constant such that This proof is complete.
From [12], for , the weighted modulus of continuity of is defined by where is a continuously differentiable function on , , and Now, we define following sequence of positive linear operators defined with the help of defined in (6):
Theorem 10 (see [12]). Let be the sequences of linear positive operators and , Ifwhere and , , and tend to zero as , then, for all functions , the inequality holds for large enough.
Theorem 11. Let be the sequences of linear positive operators defined by (27) and If , then the inequality is satisfied, where and
Proof. By simple calculation, we have From Lemma 2, we have Using Lemma 2 and (31), we obtain By means of Lemma 2 and (32), one gets Finally, from (33), we obtain If we apply Theorem 10, we obtain the desired result.
4. -Statistical Convergence
Now, let , , be an infinite summability matrix. For a given sequence , the -transform of , denoted by , is given by which provides the converging series for each We say that is a regular if whenever A sequence is called -statistically convergent to if, for every , This limit is denoted by Replacing by , the Cesàro matrix of order one in (6), from -statistical convergence, is reduced to statistical convergence. Similarly, if we take , the identity matrix, then -statistical convergence coincides with the ordinary convergence. Kolk [13] proved that, in the case of , -statistical convergence is stronger than ordinary convergence.
Further, we will first obtain the following weighted Korovkin theorem via -statistical convergence.
Theorem 12. Let be a nonnegative regular infinite summability matrix and Let be a continuous function such that Then, for all , we have
Proof. From [14], for any , it is enough to prove that So, by Lemma 2, we can easily get Again, by using Lemma 2, we haveNow, for every given , let us define the following sets: It is clear that Hence, for all , we get Therefore,
Similarly, we haveNow, we define the following sets: In view of (46), it is clear that , which yields Thus, we get
Similarly, from Lemma 3, we have Now, we give a Voronovskaja type relation for the operators
Theorem 13. Let be a nonnegative regular infinite summability matrix. Then, for every such that , one has uniformly with respect to ,
Proof. Let , and Define the function by Then, by assumption, we get and
By the linearity of applied to the last equality, we obtainIn view of Lemma 2, we get For the last term of the right hand side in (52), using Cauchy-Schwarz inequality, we are led toWe observe that , and In view of Theorem 4,Then, it follows from (32) thatuniformly in Combining (27), (31), and (46), we get the desired result.
Let denote the space of all real-valued bounded and uniformly continuous functions on , endowed with the norm Further, the appropriate Peetre -functional is given by where . By [15, Theorem ] we can find a constant such thatwhere is the second-order modulus of continuity of Now, we obtain the rate of -statistical convergence for the operators with the help of Peetre’s -functional.
Theorem 14. Let Then, one has
Proof. By the linearity property of operators and using the local Taylor formula to the function , we havewhere
Thus, we get In view of (49) for , we haveFrom (61), one can write the following: Hence, taking the limit as , we get the desired result.
The following theorem contains quantitative estimates by means of Peetre’s -functional.
Theorem 15. Letting , one has the estimate where
Proof. Letting , by (61), we have Using the last inequality for , and , we getTaking the infimum on the right sides of the above inequality for all functions , so By using the relation between Peetre’s -functional and the second modulus of smoothness given in [15], we get From (49), we get , and hence Therefore, we get the rate of -statistical convergence of the sequence to in the space
Conflicts of Interest
The authors declare that they have no conflicts of interest.